Nonlinear Hybrid Simulation of Internal
Kink with Beam Ion Effects in DIII-D
Wei Shen1, G. Y. Fu2,
Benjamin Tobias2, Michael Van Zeeland3,
Feng Wang4, and Zheng-Mao Sheng1
1Institute
for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, China
2Princeton
3General
4School
Plasma Physics Laboratory, Princeton, NJ 08543, USA
Atomics, San Diego, California 92186-5608, USA
of Physics and Optoelectronic Engineering, Dalian University of Technology, Dalian 116024, China
Outline
• Introduction
• M3D-K model and basic parameters
• MHD simulation results
• Simulations with beam ion effects
• Conclusion
Introduction
3D structure in magnetically confined fusion devices: ‘snakes’ at JET, long-
lived modes (LLMs) in the Mega-Ampere Spherical Tokamak (MAST). The
snakes, LLMs, etc, as contended by Cooper et al.[1], represent the same
physical phenomenon: saturated dominantly m=1, n=1 internal kink
modes.
In this work, the global kinetic-magnetohydrodynamic (MHD) hybrid code
M3D-K[2,3] has been applied to investigate the dynamics of the n=1 kink
mode in DIII-D sawteething plasmas.
[1] W. A. Cooper, et al, Nucl. Fusion 53, 073021 (2013).
[2] W. Park, et al, Phys. Plasmas 6, 1796 (1999).
[3] G. Y. Fu, et al, Phys. Plasmas 13, 052517 (2006).
Models used in M3D/M3D-K
G. Y. Fu, et al., Phys. Plasmas 13, 052517 (2006).
Spectrogram from localized ECE (Electron cyclotron emission) measurement near the q=1 surface contains
multiple m=1, n=1 oscillations between successive sawtooth crash events (solid vertical lines), including
fishbone-like modes that chirp down in frequency from approximately 15.5kHz to 14kHz (A). An off-axis
neutral beam source is added to on-axis neutral beam injection at t=3000ms. As the beam particle
distribution becomes more broad with time, the fishbone relaxes to a constant frequency oscillation at
around 14kHz, growing in amplitude over time until the onset of sawtooth reconnection (B).
Basic parameters
MHD simulation: linear results
U
The n=1 mode in the MHD limit, i.e.,
the beam ions are described by MHD
model and the kinetic effects of beam
ions are neglected.
Ideally unstable kink mode with linear
growth rate γ τ A=0.0141, this mode is
mainly located inside the q=1 rational
surface with dominant mode number
n=m=1.
MHD simulation: nonlinear results
As shown in the figure, the time evolution of the central pressure P(0) and the kinetic energy of different toroidal modes
indicates that the mode reaches a 3D quasi-steady state after the initial sawtooth crash, with the n=1 mode being the
dominant component.
The corresponding Poincare plots of magnetic surfaces during the saturated phase are
almost stationary, and they are shown in the left figure. The pressure profile at the
same toroidal plane is shown in the right figure, it is flat inside the q=1 surface and
consistent with the structure of magnetic surfaces.
MHD nonlinear results:
different resistivity effects
To investigate the dependence of the 3D quasi-steady state (or saturated kink) on the
resistivity, we have performed the simulations with fixed ratio of the resistivity,
viscosity and perpendicular thermal conductivity (i.e.
As shown in the figure, all cases reach 3D quasi-steady states.
).
MHD nonlinear results:
different perpendicular thermal conductivity
The left figure below shows the nonlinear evolution of the kinetic energy for several values of
at fixed resistivity and viscosity. When the perpendicular thermal conductivity decreases below a
critical value, the quasi-steady state with the saturated kink mode transits from quasi-steady states
of saturated kink to sawteeth cycles, similar to the previous results of Halpern et al. (as shown in
the right figure).
F. D. Halpern, et al., Plasma Phys. Controlled Fusion 53, 015011 (2011).
MHD results summary
To summarize our MHD simulation results, we find that the nonlinear
evolution of the n=1 kink mode results in a 3D quasi-steady state
equilibrium of saturated kink for a DIII-D discharge. It should be
noted that the ratio of
used is realistic for the expected
parameter of the experiment although the resistivity values used are
much larger than the experimental value. The experimentally relevant
resistivity values are computationally prohibitive for the code used and
cannot be considered in this work.
Simulation with beam ion effects:
linear results
To study the dependence of linear stability on the beam power, the figure below shows
the mode frequency and linear growth rate of the n=1 mode as a function of beam ion
pressure fraction at the magnetic axis Pbeam,0/Ptotal,0, with the thermal pressure Pthermal
kept fixed. When the beam pressure increases, both the mode frequency and linear
growth rate become larger.
As shown in Fig. (a), the mode structure is up-down symmetric with zero mode
frequency in the MHD limit. When the beam pressure is sufficient large, the mode
transits from a MHD-like mode to fishbone-like mode with a finite mode frequency
and twisted mode structure, as shown in Fig. (b) and (c).
With the same Pthermal and integrated beam pressure
the mode frequency and linear growth
rate decreases when the radial profile of the beam pressure becomes broader, as shown in the figure below.
The calculated mode frequencies in these figures correspond to frequencies of a few kHz consistent with
experimental measurement. Furthermore, the simulated dependence of mode frequency on beam ion profile
agree qualitatively with the measured trend of fishbone excitation. In the experiment, the fishbone tends to
be excited with higher beam power and narrower beam radial profile (i.e., on-axis heating).
Simulation with beam ion effects:
nonlinear results
Two radial profiles of the beam ion pressure with the same Pthermal and
are chosen for
the nonlinear simulation. The figures below show that the mode also saturates as a 3D quasi-steady
state after the initial sawtooth crash for both cases, with the n=1 mode being the dominant one.
Compared with the corresponding MHD results, The Poincare plots of magnetic
surfaces during the saturated phase are similar. The only difference is that, with
energetic beam ions, the structure of magnetic surfaces rotates with a finite frequency.
Similarly, the corresponding pressure profiles at the same plane (shown in Figure (b)
and Figure(c)) are flat inside the q = 1 surface and consistent with the structure of
magnetic surfaces.
To investigate whether the saturation of the
mode depends on the nonlinearity of energetic
particles or MHD, The MHD response from the
thermal plasmas is constrained to be linear by
keeping only the n = 1 toroidal perturbation. As
shown in the figure, the n = 1 kinetic energy
grows to a very large amplitude and does not
saturate. This indicates that the saturation of
the mode is due to MHD nonlinearity. Our
results are different from typical fishbone
results, in which the mode saturation is mainly
due to nonlinear flattening of the energetic
particle distribution function[G. Y. Fu, et al,
Phys. Plasmas 13, 052517 (2006)].
The n = 0 toroidal velocity is much smaller than the n = 0 poloidal velocity. Averaged poloidal zonal
velocity at the mode location is vp,ZF ∼ 2×10−3 (avA/R0), and the radius of the mode location is rmode ∼
0.2a. The frequency of the mode induced by the poloidal zonal velocity is estimated as ωZF = vp,ZF /rmode
∼ 0.01 ωA., consistent with the mode frequency in the nonlinear phase 0.006 ωA. Similar analysis could
be applied for the narrower beam profile case. In conclusion, the mode rotation in the nonlinear phase
is due to the zonal flow induced by fluid nonlinearity.
We now discuss the dependence of mode frequency on beam ion pressure profile. The
figure below shows the evolution of the mode frequency for the two beam profiles. For
the broader beam profile case, the mode frequency in the nonlinear phase is slightly
lower than the initial linear frequency. However, for the narrower beam profile case,
the mode frequency chirps down more significantly in the nonlinear phase.
Finally we discuss the effects of the n=1 mode on the beam ion profile. The figures below show the
redistribution of beam ions with v=0.705 vA due to the fishbone-like mode with
and
Pbeam,0/Ptotal,0=0.418. Note that the horizontal axis is the toroidal angular momentum and can be
regarded as a radial variable. First, after the initial sawtooth crash (at
), both of the co-
passing and trapped particles are strongly redistributed inside the sawtooth region. Then, during the
nonlinear saturation of the kink mode (from
to
), the distribution of both
co-passing and trapped particles becomes more flattened inside the q=1 surface.
Summary of results with beam ion effects
To summarize the nonlinear results with beam ion kinetic effects, we find
that the n=1 mode with beam ion effects also leads to a quasi-steady state of
saturated kink as in the MHD case in the last section. The main difference
with the MHD results is that the 3D saturated kink mode now acquires a
finite mode frequency due to the kinetic effects of beam ions. Also the mode
frequency chirps down significantly during the nonlinear evolution for the
case with a narrow beam profile. This result is consistent with the experiment
observation that the so-called fishbone-like instability tends to appear with
on-axis NBI injection where the beam profile is peaked near the axis.
Conclusion
In conclusion, nonlinear simulations of the n=1 kink mode have
been carried out with or without beam ion kinetic effects using
the kinetic-MHD code M3D-K for the parameters and profiles of a
DIII-D sawteething discharge. The simulation results show that the
n=1 kink/fishbone instability transits to a 3D quasi-steady state
after an initial sawtooth crash. With beam ion kinetic effects, the
saturated kink mode acquires a finite mode frequency on order of
a few kHz. These results agree qualitatively with the experimental
observation in the DIII-D plasmas.
Thank you for your attention